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Explicit realization of a metaplectic representation. (English) Zbl 0727.11021
Let F be a local field which is neither \({\mathbb{C}}\) nor has residual characteristic 2. Then there exists a 2-fold covering group of \(GL_ r(F)\) analogous to the metaplectic group of Weil. There exists also a class of remarkable representations of this group called the \(\Theta\)- representations. These have been constructed as subquotients of principal series representations. The purpose of this paper is to give a realization of them on a space of functions for the case \(r=3\). This involves giving a special linear form (let us call it L) on each of the representations and determining the space spanned by the functions \(g\mapsto L(gv)\) as v runs through the space of the representation. In this case the authors construct the representation on a space of functions on a 2-fold cover of \(F^ 2\)-\(\{\) \(0\}\). The existence of the model is deduced from the structure of the space of Whittaker functionals on \(\Theta\)-representations of the 2-fold covers of \(GL_ 2(F)\) and \(GL_ 3(F)\) by means of the Bernstein-Zelevinski extension of Gelfand- Kazhdan theory.

11F70 Representation-theoretic methods; automorphic representations over local and global fields
22E50 Representations of Lie and linear algebraic groups over local fields
11F37 Forms of half-integer weight; nonholomorphic modular forms
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